Why is the first one only a chopper/grinder and the second a full fledged food processor? If you know the two, can you recommend on either one? The elite one looks more promising, but I am worried it's not as a food processor as the second is. Thanks

They both have the same function. The "chop" is a function where the blades turn to 'chop'..the "grind" is a button that reverses the blades using the grinding bits on the back of the blade.
The "elitie" one might be available in more colors and sleeker form factor.
Either one would be fine and basically the same. Prehaps one does not have the blade lock function, (my mini-prep does not--but it's not really needed).
I love mine and it's used far more often than my larger one. This one sits on the counter, the larger one under the countertop. Neither of those model come with shreading disks for cheeses etc...but are not missed by me with the fact it's a 'top' counter device.

To give you some more feed back for the use of a small food processor for one or two people:
Grinding a chicken breast to make a burger, or grinding a steak make a burger for two. Cut up the meat into cube and grind, reverse..do it in batches one a fine grind, and another a courser grind.

Believe it or not. It can make a great pizza dough for one pizza. Scale back porotions, this is what I use.
1 Cup of King Arther bread flour.
1/2 tsp each of yeast, sugar and salt.
Pulse to mix.
1 Tablespoon Olive oil.
Pulse to mix.
1/3 cup of warm water. Add half mix, add more to the holes in the top (even behyond the 1/3 starting point, just a few drops) until it forms a ball.
Rest the dough 4 hour or better overnight in the 'fridge.

It's also my "Go to" for making chimichurri sauce, dressings, pesto, or salsa (just to grind onions and peppers together and a quick pulse for another addition of tomatoes)

You'll be happy with either model. Just work in small batches if you're grinding meats to burgers.

I think you'll be happy with it. I love my Mini-Prep. I have a larger food processor but it's a under-counter dweller.
this is used all the time. One application is chicken breast burgers, seasoned with lime zest, green onion, egg white, bread crumbs, a touch of soy or fish sauce. served on english muffins with a lime/olive oil slaw, and wasabi mayo and sirachi sauce. Also it's a great way to extend that 3 pack of steak or chicken breast when you just have one and need to feed two.